C₂×C₅×H(3,𝔽₃) #63203

order	270
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	Aut(G) = C₄×AGL(2,𝔽₃) (order 1728) · Inn(G) = C₃² · Out(G) = C₄×GL(2,𝔽₃)
homology	Abelianization (H₁) = C₂×C₃²×C₅ · Schur multiplier (H₂) = C₃²

Aliases

direct product	C₂×C₅×H(3,𝔽₃)
unambiguous semidirect product	(C₂×C₃²)⋊(C₃×C₅)
unambiguous semidirect product	(C₂×C₃²×C₅)⋊C₃

	subgroup	G/H
center, socle	C₂×C₃×C₅	C₃²
commutator, Frattini, cosocle	C₃	C₂×C₃²×C₅
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₅ ◁ G

44 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 90

A minimal faithful permutation representation — 2 generators on 16 points:

(1,2)(3,4,5,6,7)(9,11,14)(12,16,15);(8,9,12)(10,11,15)(13,14,16)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(270, 21);